SOLUTION
This problem is similar to the problem “XYZ=4000” posted on 11/5/2014, nevertheless a different solution is offered here.

Let be a positive integer and its prime factorization, then the number of divisors of equals . For example,
Number of divisors
Divisors

Number of divisors
Divisors

We multiply the divisors to find the triples and use multiplication tables to show the products. multiplication table
The diagonal gives the triples; the diagonal gives the triples; the diagonal gives the triples, and so on. Beware that there will be duplicates. For simplicity, we list the triples in increasing numerical order. yields distinct triples

The complete list of the distinct triples is shown below
The problem asks for ordered triples. give ordered triples.
The remaining triples give ordered triples.
Total number of ordered triples such that